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arXiv · 1508.00676

Rank gain of Jacobians over finite Galois extensions

Bo-Hae Im·Erik Wallace
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Abstract

Let $\mathcal{X}$ be a Riemann surface of genus $g>0$ defined over a number field $K$ which is a degree $d$-covering of $\mathbb{P}^1_K$. In this paper we show the existence of infinitely many linearly disjoint degree $d$-extensions $L/K$ over which the Jacobian of $\mathcal{X}$ gains rank. In the case where 0, 1 and $\infty$ are the only branch points, and there is an automorphism $σ$ of $\mathcal{X}$ which cyclically permutes these branch points, we obtain the same result for the Jacobian of $\mathcal{X}/σ$. In particular if $\mathcal{X}$ is the Klein quartic, then the construction provides an elliptic curve which gains rank over infinitely many degree $7$-extensions of $\mathbb{Q}$. As an application, we show the existence of infinitely many elliptic curves that gain rank over infinitely many cyclic cubic extensions of $\mathbb{Q}$.

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Bo-Hae Im, Erik Wallace. 2016-12-09. Rank gain of Jacobians over finite Galois extensions. https://arxiv.org/abs/1508.00676

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